Radial measurement of hydrodynamically generated concentration

A Microscale-Molecular Weight Sensor: Probing Molecular Diffusion between Adjacent Laminar Flows by Refractive Index Gradient Detection. Colin D. Cost...
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Anal. Chem. 1902, 64, 2130-2137

2190

Radial Measurement of Hydrodynamically Generated Concentration Profiles for Molecular Weight Determination Veeravagu Murugaiah and Robert E. Synovec* Department of Chemistry, BG-IO, University of Washington, Seattle, Washington 98195

Hydrodynamlcaiygeneratedconcentratkngradbnts,obtalned by InJectlnga plug of sample In a stralght cyllndrlcaltube, are manlpulatedby changlng flow rate and tube length. The ratlo of axlal veloclty to radlal veloclty experkncedby the analyte due to the comblned effect of dMurlon and convection In the flow through tube dlctates the shape of the concentratlon proflle. The radlal concentratlon gradlent Is senrltlvely measured by a refractlve Index gradlent (RIG) detector. The data are convertedInto an asymmetry ratk which Iscorrelated to the dMurlon coefflclent of the analyte and hence to the molecular welght. The effect of flow rate and tube length were extensively studled, ultknately, to determine the molecular welght of poly(ethykne glycols) (PEGS) ranglng In molecular welght from 200 to 8000 g mol-‘. The range of molecular welght that could be determlned wlth a given set of experknental condltlons was determined wlth a relatlve uncertalnty In molecular welght of 4 % for one trlal. The 8onstlMty of the RIG slgnal dependeduponthe relatlvepoldtkn of the narrow, colllmated probe beam wlth respect to the center of a flow cell. Hence, the dynamlc range of molecular welght sensing of PEGs can be selected by changlng the porltlon of the probe beam, flow rate, or tube length. Measurement of the axlal concentratlon proflle as a functlon of radlal porHlon was demonstrated uslng the RIG detector.

INTRODUCTION The characterization of water-soluble polymers such as poly(ethy1ene glycols) (PEGs) has become an activity of analytical importance because of their steadily increasing applications.’-3 Viscometry,4size-exclusionchromatography? nephelometry: and titrimetric methods7 are some of the conventional methods for the determination of PEG molecular weight. However, these methods have an inherent disadvantage of being time-consuming and require extensive sample preparation.*-7 A rapid, accurate, and precise method is needed for the analysis of PEGS and related polymers in the process environment. ~

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(1) Harris, J. M.; Dust, J. M.; McGill, R. A.; Harris, P. A.; Edgell, M. J.; Sedaghat-Herati, R. M.; Karr, L. J.; Donnely, D. L. Water Soluble Polymers: Synthesis, Solution Properties, and Applications; ACS Symposium Series 467; American Chemical Society: Washington, DC, 1991; pp 418-429. (2) Bekturov,E. A.; Bekturov,Z. Zh. Synthetic Water-SolublePolymers in Solution; Huethig & Wept Basel, 1986. (3) Molyneux, P. Water-SolubleSyntheticPolymers: Properties and Behauior; CRC Press: Boca Raton, FL, 1984; Vols. I and 11. (4) Einaga, Y.; Miyaki, Y.; Fujita, F. J.Polym. Sci., Polym. Phys. Ed. 1979,17, 2103. (5) Yau, W. W.; Kirkland, J. J.; Bly, D. D. Modern Size-Exclwion Liquid Chromatography: Practice of Gel Permeation and Gel Filtration Chromatography; Wiley: New York, 1979. (6) Cole, S.C.; Nielson, E.; Olson, W. P.; Groves, M. J.; Bassett, J. B. Anal. Chem. 1986,58, 256. (7) Price, G. F.Techniques of End-Group Analysis. In Techniques of Polymerization; Allen, P. W., Ed.; Butterworth London, 1959.

Flow injection analysis (FIA)6 is a potential solution for this problem. The physical process of solute dispersion, upon injection of a sample plug into a flowing stream, is due to the hydrodynamic processes of diffusion and convection. Dispersion of a solute plug by Poiseuille flow through a straight tube of circular cross section was first analyzed theoretically and experimentally verified by Taylor9 who solved for the cross-sectional mean concentration in the asymptotic case of long tubes at high Peclet number restriction. Following Taylor’s work several analytical and numerical simualtions have been proposed to study solute dispersion in laminar fl0w.10-14 All of these models provide mathematical tools for extracting analytical information from time-dependent measurements performed on the dispersed solute plug. Both axial (along flow) and radial (orthogonal to flow) concentration gradients of an analyte are generated when a sample in solution is injected into a flowing stream with Poiseuille flow. The presence of both axial and radial concentration gradients in a Z-configuration flow cell results in a refractive index gradient (RIG) which produces a ‘dynamic lens”. The refractive index artifact of the dynamic lens on absorbance detection was experimentally evaluated by McGuffin et al.lS Qualitative description of the RIG detection mechanism was made by Betteridge et al.16 Pawliszyn17provided a more detailed description of a RIG detector for high-performance liquid chromatography (HPLC) where the RIG along the axis of a flow in the flow cell was probed by a beam orthogonalto the direction of the flow. A significant advantage of the RIG measurement is that it can reject longterm refractive index changes such as those caused by temperature changes or gradient elution of an analyte. The RIG detector that carefully probes the radial concentration gradient at a selected radial position is under development.16-24 The RIG detector has been applied in both transient mobilephase gradient and thermal gradient microbore liquid chro~~

(8) Ruzicka, J.; Hansen, E. H. Flow Injection Analysis; Wiley: New York, 1988. (9) Taylor, G. Proc. R. SOC.London 1968, A219, 186. (10) Golay, M. J. E. In Gas Chromatography; Lansing Symposium 1957;Coates, V. J.,Noebels, H. J., Fargerson, I. D., Eds.;Academic Press: New York, 1958; pp 1-13. (11) Gill, W. N.; Ananthakrishnan, V. MChE J. 1967, 13, 801. (12) Vanderslice, J. T.; Stewart, K. K.; Rosenfeld, A. G.; Higgs, D. J. Talanta 1981,28, 11. (13) Vanderslice, J. T.; Rosenfeld, A. G.; Beecher, G. R. Anal. Chim. Acta 1986, 179, 119. (14) Mayock, K. P.; Tarbell, J. M.; Duda, J. L. Sep. Sci. Technol. 1980, 15, 1285-1296. (15) Evans, C. E.; Shabushnig, J. C.; McGuffin, V. L. J. Chromatogr. 1988,459, 119-138. (16) Betteridge, D.; Dagless, E. L.; Fields, B.; Graves, N. F. Analyst 1978,103,897. (17) Pawliszyn, J. Anal. Chem. 1986,58, 243-246. Renn, C. N.; Synovec, R. E. Anal. Chem. 1990, (18) Hancock, D. 0.; 62, 2441-2447. (19) Hancock, D. 0.; Synovec, R. E. Anal. Chem. 1988,60,1915-1920. Synovec, R. E. Anal. Chem. 1988,60,2812-2818. (20) Hancock, D.0.; Synovec, R. E. J. Chromatogr. 1989,464,83-91. (21) Hancock, D. 0.; (22) Renn, C. N.; Synovec, R. E. J . Chromatogr. 1991,536, 289-301. (23) Murugaiah, V.; Synovec, R. E. Anal. Chim. Acta 1991,246,241249. (24) Murugaiah, V.; Synovec, R. E. SOQUE Lasers ’90Conf. Proc., Lasers Chem. 1991, 13, 763-771.

0003-2700/92/0364-2130$03.00/0 0 1992 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992

matography.22 The measurement of the RIG signal arising as a consequence of a concentration gradient is a suitable approach to measure D, and hence the molecular weight of water-soluble polymers such as PEGSwhich have no suitable chromophore for conventional absorbance detection. Molecular weight and the diffusion coefficient, D,, have been measured by absorbance detection using FIA meth0d~.12.13~26,26 These FIA-based methods to determine molecular weight12J3925926 have been limited to samples that have suitable absorbance spectra. Other techniques such as capillary zone electrophoresis (CZE)z7in which the signal is a function of D, can be used to determine molecular weight. Techniques involving static systems as well as the frontal analysis,reviewed by Parcher,a which is often used to obtain physicochemical parameters about the analytes, are time consuming. In this study, we will first explain why radial measurement of the concentration gradient is superior in terms of signal sensitivity and molecular weight sensitivity. We have examined the role of diffusion and convection in determining the asymmetry of the signal due to the radial concentration gradient in a flow-through tube. Consequently,the flow rate and the flow-through tube length were manipulated to select the suitable experimental conditions to determine the molecular weight of PEGS in a selected range of molecular weights. Although both the tube and flow cell contributes to the concentration profile broadening, only the tube length can be readily modified. Thus, only the dependence on tube length was examined. A careful manipulation of these parameters has led to the development of a tunable molecular weight analyzer which is reported. Molecular weight sensitivity of the analyzer for different ranges of molecular weights was critically examined. Secondly, the analytical sensitivity of radial concentration gradients at preselected radial positions, under the same conditions of flow rate and tube length, was examined. Spatial concentration profile information which encodes the Poiseuille flow characteristics across the section of the flow through tube or flow cell obtained in this manner cannot be achieved by techniques which rely on axial concentration gradient measurement or that involve plug flow such as CZE.27 Thirdly, an important feature of the selectiveradial probing allowed us to simultaneously make measurements of a wide range of molecular weights. The method described here is inexpensive, fast, and readily amenable for on-line analysis.

THEORY It is important to evaluate the means of optically probing the concentration profile encountered in FIA and HPLC. Figure 1 illustrates two ways of probing the concentration versus time profile in a flow-through tube. Probe beam a, which is orthogonal to the flow (axialmeasurement),measures the average concentration across the section that the probe beam passes. The area averaged concentration at any time, t , is given by

where R is the radius of the tube, and C ( t ) ,is the concentration at a radial position r away from the axis.29 In this measurement, the information on the radial concentration distribution (25) Gerhardt, G.; Adams, R. N. Anal. Chem. 1982,54, 2618. (26) Trumbore, C. N.; Grehlingei, M.; Stowe, M.; Kelleher, F. M. J. Chromatogr. 1986,322,443. (27) Walbroehl, Y.; Jorgensen, J. W. J . Microcolumn Sep. 1989,1,41. (28) Parcher, J. F.Adv. Chromatogr. 1978,16, 151. (29)Reijn, J. M.; Van der Linden, W. E.; Poppe, H. Anal. Chim. Acta 1980,114,105-118.

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Time Flgure 1. Comparison of the opticelmeasurementof the concentration versus time profile of an analyte InJected Into a Polsuellle flow by probe beams orthogonal and parallel to the direction of flow. Probe beam a measures the axial concentratlon profile resulting In a proflle wlth peak width W,. Integratlonof the radial concentrationgradient proflle, meesured by refractbe Index gradient detectlon, results In the concentration proflle at a dlstance r from the center axls and has a bandwidth W,, smaller than W..

of the analyte is lost due to the averaging effect, and the concentration profile obtained by probe beam a results in a broad bandwidth, W,,as shown in Figure 1. Mayock et al.14 reported simulations of the concentration profile at three different radial positions that were not verified by experiments. If the probe beam is wide, it is difficult to make direct measurement of a concentrationprofile at a radial position, C(t),, by probing with a beam parallel to the axis of the flow, probe beam b, as shown in Figure 1. However,Wightman et al. measured the radial concentration profile with a microelectrode to obtain unique information on the bandbroadening effects in FIA and HPLC.30 It is our intent to demonstrate that C(t), can be optically measured with a narrow collimated beam, with the resultant data useful in molecular weight determinations. Further, a wide range of molecular weight calibrations can be obtained by probing at selected radial positions, described later. This provides second-order information which is not available in simple hydrodynamic flow with axial measurements." The angular deflection, 8, in the RIG measurement for the Z-configuration flow cell by probing with a beam parallel to the axis is given by18 8 = (L/n) dC(r)ldr (dnldC) (2) where L is the path length of the beam orthogonal to the radial concentration gradient dC(r)/dr,r is the radial position of the probe beam, n is the nominal refractive index (RI) of (30) Wightman, R.M.; May, L. J.; Baur,J.; David, L.; Kristeneen, E. ACS Symp. Ser. 1989, No.403 (Chem. Sens. Microinstrum.), 114-128.

Taylor’sgwork,the time, t, required for sufficient radial mixing to achieve a Gaussian peak is

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Thus, for D, = 10-7 cm2 s-1 (large polymer) and a tube radius of 0.01 cm, the time required to achieve a Gaussian peak and apply the Golay equation is 70 s and an analysis time of approximately 2t, or 140 s, is too long. Also, note that calibration of different molecular weight ranges is not possible with Gaussian peaks due to lack of radial concentration gradient asymmetry, as explained later. All of these methods, though, have fairly comparable molecular weight sensitivity in the molecular weight determination; the reduced analysis time of our method is the discriminating factor.

ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, l9Q2 2133

velocity is not actually a velocity in the usual sense, but a mass transport mechanism that leads to an attenuation of the radial concentration gradient.34 In analogy with the Einstein equation, radial diffusion is characterized by d2 = 2D,t (8) where d is the characteristic diffusion length. Equation 8 can be written as u, = d l t

2DJR

(9) where dlt is a measure of effective radial velocity, ur,and R is the radius of either the flow cell or tube. Since the axial velocity at a given radial position at a distance r from the center of the flow cell or tube, u,(r), is proportional to the volumetric flow rate, F, a ratio of radial velocity to the axial velocity would incorporate the effect of diffusion and convection. Hence a

u,Iu,(r) a 2D,lRu,(r) (10) would empirically determine the shape of the concentration profile. Technically, eq 10 is a modified form of the Golay equation in terms of the peak-broadening mechanisms. Therefore, for a given length of flow-through tube, the ratio u,lu,(r) is the dominant factor to predict the shape of the concentration profile, as depicted in Figure 1. For a large ur/ua(r)ratio, Le., for small molecules or low flow rates, the concentration profile would approach a Gaussian shape as described by the Golay equation3I and the AR for the RIG signal would approach unity. Conversely, for a small u,Iu&) ratio, the concentration profile would result in a sharp rise and exponential decay, resulting in a high AR for the RIG signal.13 Combining eqs 6, 7 , and 10, one obtains

(D,lflk6 a (M,lflka (11) which indicates the relations between AR, D,, and M,, and F and the coefficientsk5 and k6 may be obtained by logarithmic plots of AR versus D , and molecular weight, respectively. Equation 11 will be experimentally evaluated. The concentration profile and hence the AR of the RIG signal is also dependent on the length of the flow-through tube, L. As pointed out by Golay et al. in another report,35 the variances contributed by short open tubes, connected to one another, are not additive unless there is a complete radial mixing at the connections, which is not desirable in our experiments. Further, there is no explicit closed-form solution to demonstrate the effect of the tube length and flow cell combination on the concentration profile for such short tubes. A simple expression describing the fractional radial diffusion distance in a tube will allow one to consider not only the D , and F dependence but also tube length L dependence. Starting with eq 8, one can derive an expression for the fractional diffusion distance, dlR, with AR

a

( d / R ) 2= 2.rrDmL/F (12) While u,/u,(r)in eq 10 should depend only upon a given analyte D , and flow rate F and be independent of tube length L, d / R will provide a more comprehensive description of the radial concentration gradient dependence on D,, L, and F.

Next, let us consider how different molecular weight calibration curves (and ranges) can be examined a t a constant flow rate, with probing a t different radial positions in the (34) Giddings, 3. C. Dynamics of Chromatography; M. Dekker: New York, 1965. (35) Atwood, J. G.; Golay, M. J. E.J . Chromatogr. 1981,218,97-122.

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Figure 3. (A, Top) schematic diagram of the experimental setup: L, He-Ne Laser; MO, microscope obJectlveand fiber chuck on precislon fiber coupler: F,single-mode fiber; GL,GRIN lens, FC, Zconfiguratlon flow cell; W, waste; PSD, posltlon-sensltlve detector; C, computer; CR, chart recorder: V, InJectlonvalve; P, syringe pump. (6, bottom) flow cell configurationwlth respect to the laser probe beam: L, laser beam; R, radius of the flow cell; DB, deflected beam; r, radial distance of the probe beam; 0 = angle of deflection; I, solvent Inlet; W, solvent outlet: X, horizontal center axis of the flow cell.

flow cell. The velocity of the individual streamlines in a parabolic profile is given by u,(r) = 2 ( u , ) ( l - r2/R2) (13) where (u,) is the mean axial velocity and u,(r) is the individual streamline velocity at a distance r from the center axis of the tube. According to eq 10, the shape of the concentration profile depends on the velocity experienced by the molecules in an individual streamline. Hence, from eqs 6,7,10,and 13 different asymmetric concentration profiles can be obtained from probing the dispersed analyte a t different radial positions in the flow cell using a narrow collimated beam, since

AR a (D,lu,(r))ksa (M,/u,(r))ks (14) enabling one to make multiple molecular weight calibrations under the same experimental conditions of flow rate and tube length. This is made possible by probing the radial concentration profile, since the axial concentration gradient measurement for either Poiseuille or plug flow (CZE)",2*does not provide this spatial information, as well as probing for radially symmetric Gaussian peaks. EXPERIMENTAL SECTION A block diagram of the apparatus used in this study is shown in Figure 3A. The 633-nm, 5-mW continuous-wave (cw) output from a He-Ne laser (Model 105-1,Uniphase, San Jose, CA) waa focused by a 2 0 microscope ~ objective (NewportCorp., Fountian Valley, CA) onto a single-modeoptical fiber (4-pm core, 125-pm clad, 250-pm jacket) (Newport Corp., Fountain Valley, CA) which

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Table I. Physical Examined ~ , , a 10-7~,? g mol-' cm2 s-1 200 67.8 297 54.56 387 47.17 582 37.68 889 29.85

64,

NO. 18, SEPTEMBER 15, 1992

Constants for Poly(ethy1ene glycols) labelc PEG200 PEG300 PEG400 PEG600 PEG900

M,,"

g mol-'

1433 3199 4611 8239

10-7~,,b om2s-I 22.96 14.76 12.07 8.77

label' PEG1450 PEG3350 PEG4600 PEG8000

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Weight average molecular weight, M,. Diffusion coefficient, D,, for poly(ethy1ene glycols),in water at 25 O C , according to eq 15. Nominal molecular weight label used for discussions in the text.

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is optimized for a wavelength of 633 nm. The optical fiber was mounted on a precision fiber coupler with a fiber chuck (Newport Corp., Fountain Valley, CA). A collimated probe beam was produced by interfacing the optical fiber output to a GRIN lens (SELFOClens, NSG America,Sommerset,NJ) that was quarter pitch at 633 nm. The narrow, collimated probe beam, 200 pm in diameter at the flow cell entrance, was passed through a Zconfiguration flow cell, made in-house, with a 10-cm distance separating the flow cell and the GRIN lens. The flow cell had a 1.0-cmlength and an 800-pm diameter with l/l&.-o.d. X 0.007in4.d. PEEK tubing (Upchurch ScientificInc., Oak Harbor, WA). The flow cell was mounted on a high-precision X-Y-2 translational stage (Newport Corp., Fountain Valley, CA) which allowed radial adjustment of the position of the beam (Figure 3B) in order to readily examine the radial dependence of the hydrodynamically generated concentration gradient. After the collimated laser beam passed through the flow cell, it was sent onto a bicell position-sensitivedetector (PSD) (Hamamatsu City, Japan). The optical and mathematical configuration of the PSD device was discussed in detail elsewhere.21 The ratio output of the PSD is independent of the incident beam intensity. The beam deflection, 8,due to the RIG was proportional to the PSD ratio output. Deionized water served as the carrier and was delivered by a syringe pump (ISCO, Model 2600, Lincoln, NE) through an automated injection valve fitted with a 2.5-pL injection loop and a two-position electric actuator (Valco Instruments Co. Inc., Houston, TX). Since there can be segregation of PEG, due to polydispersity, semisolid PEG was melted in a hot water bath and flake sample was crushed and well mixed to get a representative sample for weighing. Approximately 200 ppm aqueous solutions of all PEGS listed in Table I were prepared from stock solutions of 5000 ppm. For studies of the AR dependence on the flow rate and tube length, a 50-cm length of 0.007-in.4.d. PEEK tubing was connected between the injection valve and the flow cell. The laser probe beam was set at the optimum position,'* r / R = 0.58 (Figure 3B), by carefully positioning the flow cell X-Y-Z translational stage vertically relative to the probe beam. Flow rates ranging from 50 to 150 pL/min at an interval of 25 pL/min were examined with five replicate injections made for each PEG standard. The RIG signals were recorded on a chart recorder and collected via a laboratory interface board (IBM Data Acquisition and Control Adapter, IBM, Boca Raton, FL) with a personal computer (Leading Edge, Model D, Canton, MA). Similar measurements were done with 39.4- and 27.8-cm tube lengths while keeping the tube radius, R, constant. For the studies of the radial dependence of the sensitivity of the AR, a 27.8-cm tube length and a 75-pL/min flow rate were used for all the PEGS listed in Table I. The relative position of the probe beam with respect to the center of the flow cell was carefully adjusted at three different values, r / R = 0.25,0.50, and 0.63. First, the RIG signal is base-line adjusted so that the noise was centered about zero. The RIG signal was integrated from beginning to end, point by point, to produce a concentration profile.

RESULTS AND DISCUSSION ConcentrationGradient Signal as a Function of Flow Rate, Tube Length, and PEG Diffusion Coefficient. In order to examine the concept of correlating the AR to ur/ua(r)

0.000

0.005

0.010

D m I F , (cm-') Figure 4. Asymmetry ratlo, AR, of the refractive Index gradlent signals of poly(ethy1ene glycols) listed In Table I ((A) PEG 8000, (B) PEG 200)

using a 27.&cm tube length, as a function of the ratio of the dlffuslon wlth the laser probe coefficlent (D,,,) to the volumetric flow rate (4, beam at rlR = 0.58. Upper, Lp, and lower, L,, limits for DmlFare defined In the text.

150 uUmin

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D m i F , (cm-') Figure 5. Asymmetry ratlo, AR, of the refractiveIndex gradlent slgnals of poly(ethyleneglycols) llsted In Table I ((A) PEG 8000, (B) PEG 200) uslng a 39.4-cm tube length, as a function of the ratio of the dlffusbn coefficlent (0,)to the flow rate, wlth the laser probe beam at rlR = 0.58.

and d / R , by eqs 10-12, D J F values were first calculated for the PEG standard runs with three tube lengths at the five flow rates listed in the Experimental Section. Figures 4 and 5 show the plot of AR versus D,/F, depicting the relationship given by eq 11,for two extreme cases of flow rates, 50 and 150 pL/min, with two tube lengths, 27.8 and 39.4 cm, respectively. At the low flow rate, 50 pL/min, udu,(r) is large enough so that a small analyte such as PEG 200 has sufficient time to diffuse and form almost a Gaussian profile, resulting in a low asymmetry ratio. On the other hand, when the flow rate is increased to 150 pL/min, the concentration profile becomes more asymmetric, resulting in an increase in the AR. It was observed that for the 27.8-cm tube at 50 pL/min (Figure 4), the concentration profile is approximately Gaussian and the ARfalls below 1.95when D J F is greater than 0.0045 (labeled as the LZlimit), whereas AR increases when D,/F is less than 0.0045. At 150 pL/min (Figure 4), because of the higher asymmetry in the concentration profile, the AR increases and then begins to decline due to the change in the shape of the concentration profile when D J F is below 0.0011, labeled as L1. The limits L1 and Lz will be formally defined shortly. The trend of AR changing between two limits of D,/F for a given length of tube was consistent in all 15combinations of flow rate and tube length. The range of the limits L1 and Lt

ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992

”

Table 11. Upper and Lower Limits of D,,,/F,. for Different Tube Lengths, Which Demarcate the Different Types of Concentration Profiles in a Flow-Through Tube under a Given Set of Experimental Conditions tube length, L 27.8 39.4 50.0

Lz DdF 0.0041 0.0028 0.0023

sb 0.0004 0.0002 0.0003

(dlR)f 0.85 0.83 0.85

L1 DJF 0.0010 0.0005 d

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(dIR)IC 0.42 0.35 d

For five flow rates (50,75,100,125,and 150 pLlmin) at each tube length. Units of D d F a r e cm-l. See Figures 4 and 5 for illustration of the maximum limit LZ and minimum limit L1 as well as the definition in text. Standard deviation of the respective DJF. d l R, calculated for each limit L1 and Lz following eq 12, using the tube length, L. L1 not reached for L = 50-cm tube length. Limited by the maximum molecular weight in the sample set, 8000 g mol-’. a

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50 uUmin 150 uUmin

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2000

4000

6000

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MOLECULAR WEIGHT for D,/ F for a given tube length is more important than the exponents k5 and k6 of eq 11, for predicting the concentration profile and as such exponents ks and k6 of eq 11 were not determined in this work. However, for given experimental conditions of tube length and flow rate, the exponents k5 and k6 of eq 11 were estimated in our previous work.23 From the foregoing discussions, two limits L1 and LZfor D J F values, as shown in Figures 4 and 5, can be defined to demarcate the diffusion coefficient dependent changes in the shape of the concentration profile, as reflected by the ur/ua(r) ratio. At the limits L1 and Lz, dARId(DJ0 is zero. When D J F is greater than the upper limit Lz,the concentration profile of the analyte is almost Gaussian due to the dominance of radial diffusion resulting in a relatively symmetric RIG signal. In this region, the AR bears no discriminating ability between consecutive polymers. When D,lF is between the two limits L1 and Lz,the concentration profile of the analyte is asymmetric due to the combined effect of diffusion and convection resulting in an asymmetric RIG signal. When D J F is less than the lower limit L1, the concentration profile goes through a transition region due to high asymmetry. In this region the concentration profiie is most probably governed by convection dominating the leading peak with a diffusion “shoulder” resulting in a “double hump” peak, which resulta in a decline of AR. For analytical purposes,the most sensitive correlation of AR versus DmlF is the rising region between L1and LZin Figures 4 and 5, where dARId(DJ0 is steep. As discussed in the next section, this region is of interest for this device to obtain molecular weight information. The corresponding upper and lower limits of D,/F values are given in Table 11. On examination of the mean and standard deviation values for D,lF, which covered all five flow rates, the small standard deviation values imply that the D J F limits are independent of flow rate and that ur/ua(r), when operation is over a given length L, is the underlying property of the flow system which governs the shape of a concentration profile. Further, it is clear from Table 11, that u,/u,(r),when operation is over a different tube length, yields different limits. The two limits L1 and LZare statistically different for different tube lengths. The dependency of the AR on D J F and tube length L (eq 11)leads to a means of setting up experimental criteria to tune the experimental device using the d/R limits (eq 12) in Table 11to determine molecular weights in different ranges. Constancy of the d / R values at L1 and LZcalculated for the limiting D J F values for the respective tube length, as given in Table 11, implies that eq 12 incorporates the overall effect of the flow rate, tube length, and D, of the analyteto predict the concentration profile. At the lower limit of D,/F, L1for each tube length, the relative diffusion distance, d/R in Table 11, is small enough to produce a sharp rise in concentration profile resulting in a highly asymmetric RIG signal. At the upper limit of D J F ,

Flguro 6. Callbration curve showlng the dependence of AR on molecular weight for the 39.4-cm tube at two different flow rates wlth the laser probe beam at rlR = 0.58.

LZfor each tube length, the relative diffusion distance d/R reaches a maximum value of about 0.85. With sufficient interplay of diffusion and convection, the analyte approaches a diffusion-limited maximum corresponding to limit Lz, resulting in a Gaussian profile at the exit of the tube. According to eq 12, since the flow cell is only 1cm in length, the relative diffusion distance in the flow cell is negligible compared to the tube. This would imply that one should get an AR of unity for the RIG signal. However, when you consider the detection mechanism of the RIG signal in the flow cell, though relative diffusion is negligible, the absolute diffusion distance in the flow cell contributes additional broadening, resulting in a small asymmetric peak which accounts for an AR around 1.95 under these conditions. Since variances are not additive (flow cell and tube), experimental evidence suggests that adjusting tube length effectively controls the concentration profile, although flow cell contribution to variance is significant. Precision and Range of the Molecular Weight Determination. Since the shape of the concentration profile is determined by the ur/u,(r)ratio for a given tube length, and hence the D d F value for a given experimental condition, and the AR for practical utility is bound within a range of D J F reported in Table 11, only PEGS within a molecular weight range can be determined by the selected experimental conditions. The correlation of D, to molecular weight, Ad, for PEGS, in water at 25 “C, as estimated from reference values,36 is given by

D, = 1-25 x

(15) The dependence of the AR on molecular weight using the 39.4-cm tube at the two extreme flow rates examined is given in Figure 6 (AR vs D J F shown in Figure 5). For a given length of tube, a lower flow rate is suitable for high molecular weight polymers and higher flow rate is suitable for low molecular weight polymers. The sensitivity of this technique depends upon dAR/dM, which should be large to provide a good molecular weight prediction. Ideally, one would want the slope of the rising portion in Figure 6 to be 88 large ae possible and to have an almost constant and small uncertainty in AR. From the upper and lower limits of D J F in Table 11, the lower and upper molecular weights that can be theoretically measured at different flow rates for the 39.4-cm tube are computed using eq 15 and shown in Figure 7. This gives the dynamic range of the molecular weight analyzer for the 39.4-cm tube at the five flow rates. The dynamic range also 10-4~-055

(36)Polymer Handbook, 3rd ed.; Wiley: New York,1989; Vol. 7, p 62.

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ANALYTICAL CHEMISTRY, VOL. 84, NO. 18, SEPTEMBER 15, 1992 100000

w

z

Min. Mol. Wt.

2

E

a

g

lO0Oi

100 25

50

75

100

125

150

175

0

FLOW RATE, U m i n

20

30

40

TIME, s Flgurr 8. Radlal concentration, qi), profile of PEG 1450 at different laser beam positions: (A) concentration profile for rlR = 0.25; (E) concentration profile for rlR = 0.5. Curve B is shifted vertically for clarity. The length of the tube was 27.8 cm, with a 75 pLlmin flow rate. The concentrationprofilewas obtained by integratingthe baselinecorrected RIG signal with respect to time.

Flgurr 7. Lower and upper molecular weight limits of poly(ethylene glycds) that can be determinedfor a given set of experimentalconditkns as calculated from the limb of D,lFiisted In Table 11, forthe 39.4-cm tube and laser probe beam at r/R = 0.58.

changes as a function of tube length, L, since the DJF limits change with L (Table 11). Since the sensitivity of the analyzer is not constant over a wide range of molecular weight, the molecular weight has to be taken into account to define the sensitivity of the measurement as a means of normalizing the relative uncertainty of the AR. A sensitivity parameter, S,, is defined at a certain molecular weight, Mi, as

S, = (dARldM),Mi (16) where (dAR1dM)i is the slope of AR versus M near the molecular weight Mi of interest. The precision of the AR can be related to the error associated with the signal measurements, i.e. e(+) and e(-). From the definition of the AR in eq 5, the law of error propagation gives where so(+) and so(-) are the standard deviations of e(+) and e(-), respectively. It becomes evident that the precision of the AR at any arbitrarily selected experimental condition (flow rate and tube length) is a very sensitive function of the molecular weight of the analyte. It was observed experimentally that the relative uncertainty in the measurement of e(-) is greater than that of e(+) for all measurements because of the lower signal-to-noiseratio of the trailing peak (seeFigure 2). Thus, the term [s0(-,/6(-)1 is the major contributor to the relative standard deviation of the AR. Experimentally, the standard deviation, sm, of the replicate AR measurements was approximately linear with AR and fit a linear regression equation

= 0.0456AR - 0.0285 (18) The relative molecular weight uncertainty of a determination near Mi is given by sAR

In a polymerization process control analysis, the relative molecular weight uncertainty is an important parameter. In order to achieve a premium product with a narrow molecular weight distribution during polymerization, AMrel should be kept as small as possible, and the analytical method should possess small molecular weight uncertainty. Both AMreland AM (absolute molecular weight uncertainty) should be minimized. At each flow rate for a given length of tube, the sensitivity

10

'

parameter S, defined by eq 16was calculated for all the PEGs examined. The molecular weight of polymer with optimum S, that could be most sensitively identified under a given set of conditions (flow rate and tube length) was identified but not shown for brevity. As a typical case, for the 39.4-cm tube at 50 pL/min flow rate, the optimum molecular weight was 3350, which gave an AR of 3.02,as shown in Figure 6, and an S, of 2.2. From eq 18, SAR = 0.109 and, from eq 19, = 5.0% and Ah4 = 170 g mol-'. But at 150 pllmin, PEG 400 gave an AR of 2.91 and an S, of 2.75, resulting in AMre,= 3.8% and AM = 15 g mol-'. Thus, at the higher flow rate, changes of less than a monomer unit can be sensitively measured for low molecular weight PEGs. Further, it was experimentally determined that a 6 improvement is obtained for replicate AR measurements, where n is the number of replicate injections. Thus, AMrel can be improved by 6,requiring n trials, with each trial requiring 30 s to 1min. Selective Probing of the Radial Concentration Gradient. In a previous study, the RIG signal of a given sample was measured by probing at five different rlR positions of the laser beam.24 For each RIG signal as in Figure 2, three time points were referenced the time between injection and e(+), the time taken to reach zero crossing base-line, and the time taken to reach e(-). Examination of these three times defined as above revealed a dependence on the rlR value of the probe beam; Le., all three times progressively increased with an increase in rlR value. The results demonstrated a Poiseuille flow profile in the tube and flow cell.1834 Because of the parabolic flow profile modified in the tubing and flow cell, the streamlines close to the center (small rlR for the probe beam) have greater speed and hence the time to reach e(+) is reduced more than in the case of streamlines away from the center (larger rIR). The same trend was also observed for the zero crossing time and e(-) time. Since the RIG signal is a measure of the radial concentration gradient at a particular radial position and the radial concentration gradient is a function of the axial concentration profile's at a given rlR, integration of the RIG signal should reproduce, to a good approximation, the axial concentration profile at a given rlR, illustrated as C ( t ) , in Figure 1. One should expect to see a shift in the concentration profile along the time axis and a change in the concentration profile symmetry as rlR is changed, at constant flow rate and tube length. The integrated concentration profiles for two rlR positions (0.25 and 0.50) of the probe beam using PEG 1450 as analyte are shown in Figure 8 (the original RIG signal is

ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992

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MOLECULAR WEIGHT Figure 9. Dependence of RIG slgnal asymmetry on the probe beam posltlon reflected by D,lFversus AR plots: (A, top) D,lFcalculated wlthaveragefbwratefordataobtalned byproblngattworadialpositkns, rlR = 0.25 and 0.50; (B, middle) &IFcalculated wlth adjustment for parabolicvebcky proflle according to eq 20 for data obtalned by probing at two radial posltlons,rIR = 0.25 and 0.50 (error bars of plus or mlnus one standard devlatlon shown); (C, bottom) caiibratlon of AR versus M obtalned using a 27.8tm tube at a 75 pL/mln flow rate wlth 250 ng of injected mess (calibrationobtained wlth the probe beam at r/R = 0.25 shows sensitive dlscrlminatlon of lower molecular weight PEGS, and calibratlon obtained wlth the probe beam at rlR = 0.50 Is suitable for hlgher molecular welght PEQs).

2117

shown in Figure 2 for rlR = 0.50). A similar shift in the concentration profile along the time axis was obtained for all other PEGs at all rlR positions studied. The selective measurement reported here was possible with the present optical technique of using a narrow, collimated probe beam produced by a GRIN lens, as depicted in Figure 1. In conventional absorbance detection, even though the beam is parallel to flow, the beam generally overfills the flow cell aperture, resulting in the measurement of C(t),, as shown in Figure 1. Molecular Weight Calibration by Selective Radial Probing. The preservation of the Poiseuille flow in the flow cell and the establishment of distinct concentration profiles at different radial positions result in different asymmetric RIG signals for the same analyte at different radial positions (eqs 13 and 14). As described earlier, DJF is a unifying hydrodynamic quantity that determines the concentration profile for a given tube length. A plot of AR versus D J F with proper adjustment for the linear velocity at different radial positions will reinforce this unifying dependence on DJF. AplotofDJFforthedataobtainedbyradiallyprobing at rlR = 0.25 and 0.50 is given in Figure 9A. Initially, these data are plotted at the same F,thus, the same axial linear velocity. The velocity of the individual streamlines in a parabolicprofile is given by eq 13. For the two radial positions considered here u,(r) (rlR = 0.25) = 1.25uJr) (rlR = 0.5) (20) Figure 9B gives the dependence of AR on DJF adjusted for the respective linear flow velocity according to eq 20. Note that the curves in Figure 9B at two different rlR are now overlapped within experimental uncertainty. The differences in the concentration profile at different radial positions, due to different flow velocities, can be used for improved molecular weight calibration. When a sample plug is injected into a Poiseuille flow in a tube, at a radial position further away from the center, the molecules experience smaller axial velocity than at a position close to the center. However, the radial diffusional velocity, u,,remains the same. Therefore, for a given tube length and flow rate, the u,lua(r)varies for different radial positions, influencing the shape of the concentration profile. The AR of a RIG signalvaries accordingto the concentrationprofile determined by the u,/ua(r)ratio. Thus, the use of selective radial probing has the important advantage of altering the sensitivity of the AR on molecular weight and providing selective adjustment of the molecular weight range that can be analyzed. Figure 9C shows the calibration of the AR versus molecular weight obtained at r/R equal to 0.25 and 0.50, respectively, using a 27.8-cm tube length at 75 pLlmin flow rate by injecting 2.5 pL of 100 ppm PEG solutions. The calibration at r / R = 0.25 is suitable for the determination of low molecular weight PEGs, from 200 to 3000 g mol-'. The calibration at r / R = 0.50 is suitable for the determination of higher molecular weight PEGs, from 1OOO to 8OOO g mol-'.

ACKNOWLEDGMENT We gratefully acknowledge the support of this work by Union Carbide Corporationand helpful discussionswith Jerry R. Hale. RECEIVED for review February 4, 1992. Accepted May 27, 1992. Registry No. PEG,25322-68-3.